# Runcinated 120-cell

 Orthogonal projections in H3 Coxeter plane 120-cell Runcinated 120-cell (Expanded 120-cell) Runcitruncated 120-cell 600-cell Runcitruncated 600-cell Omnitruncated 120-cell

In four-dimensional geometry, a runcinated 120-cell (or runcinated 600-cell) is a convex uniform polychoron, being a runcination (a 3rd order truncation) of the regular 120-cell.

There are 4 degrees of runcinations of the 120-cell including with permutations truncations and cantellations.

The runcinated 120-cell can be seen as an expansion applied to a regular polychoron, the 120-cell or 600-cell.

## Runcinated 120-cell

Runcinated 120-cell
Type Uniform polychoron
Uniform index 38
Coxeter-Dynkin diagram
Cells 2640 total:
120 5.5.5
720 4.4.5
1200 4.4.3
600 3.3.3
Faces 7440:
2400{3}+3600{4}+
1440{5}
Edges 7200
Vertices 2400
Vertex figure
Equilateral-triangular antipodium
Schläfli symbol t0,3{5,3,3}
Symmetry group H4, [3,3,5], order 14400
Properties convex

The runcinated 120-cell is a uniform polychoron. It has 2640 cells: 120 dodecahedra, 720 pentagonal prisms, 1200 triangular prisms, and 600 tetrahedra. Its vertex figure is a nonuniform triangular antiprism (equilateral-triangular antipodium): its bases represent a dodecahedron and a tetrahedron, and its flanks represent three triangular prisms and three pentagonal prisms.

### Alternate names

• Runcinated 120-cell / Runcinated 600-cell (Norman W. Johnson)
• Runcinated hecatonicosachoron / Runcinated hexacosichoron / Runcinated polydodecahedron / Runcinated polytetrahedron
• Small diprismatohexacosihecatonicosachoron (acronym: sidpixhi) (George Olshevsky, Jonathan Bowers)[1]

### Images

Schlegel diagram (Only tetrahedral cells shown) Orthogonal projections in Coxeter planes H3 A2/B3

## Runcitruncated 120-cell

Runcitruncated 120-cell
Type Uniform polychoron
Uniform index 43
Coxeter-Dynkin diagram
Cells 2640 total:
120 (3.10.10)

720 (4.4.10)
1200 (3.4.4)
600 (3.4.3.4)

Faces 13440:
4800{3}+7200{4}+
1440{10}
Edges 18000
Vertices 7200
Vertex figure
Irregular rectangular pyramid
Schläfli symbol t0,1,3{5,3,3}
Symmetry group H4, [3,3,5], order 14400
Properties convex

The runcitruncated 120-cell is a uniform polychoron. It contains 2640 cells: 120 truncated dodecahedra, 720 decagonal prisms, 1200 triangular prisms, and 600 cuboctahedra. Its vertex figure is an irregular rectangular pyramid, with one truncated dodecahedron, two decagonal prisms, one triangular prism, and one cuboctahedron.

### Images

Schlegel diagram (Only triangular prisms shown) Orthogonal projections in Coxeter planes H3 A2/B3

## Runcitruncated 600-cell

Runcitruncated 600-cell
Type Uniform polychoron
Uniform index 44
Coxeter-Dynkin diagram
Cells 2640 total:
120 3.4.5.4
720 4.4.5
1200 4.4.6
600 3.6.6
Faces 13440:
2400{3}+7200{4}+
1440{5}+2400{6}
Edges 18000
Vertices 7200
Vertex figure
Trapezoidal pyramid
Schläfli symbol t0,1,3{3,3,5}
Symmetry group H4, [3,3,5], order 14400
Properties convex

The runcitruncated 600-cell is a uniform polychoron. It is composed of 2640 cells: 120 rhombicosidodecahedron, 600 truncated tetrahedra, 720 pentagonal prisms, and 1200 hexagonal prisms. It has 7200 vertices, 18000 edges, and 13440 faces (2400 triangles, 7200 squares, and 2400 hexagons).

### Images

Schlegel diagram Orthogonal projections in Coxeter planes H3 A2/B3

## Omnitruncated 120-cell

Omnitruncated 120-cell
Type Uniform polychoron
Uniform index 46
Coxeter-Dynkin diagram
Cells 2640 total:
120 4.6.10
720 4.4.10
1200 4.4.6
600 4.6.6
Faces 17040 total:
10800 {4}, 4800 {6}
1440 {10}
Edges 28800
Vertices 14400
Vertex figure
Chiral scalene tetrahedron
Schläfli symbol t0,1,2,3{3,3,5}
Symmetry group H4, [3,3,5], order 14400
Properties convex

The omnitruncated 120-cell is a convex uniform polychoron, composed of 2640 cells: 120 truncated icosidodecahedra, 600 truncated octahedra, 720 decagonal prisms, and 1200 hexagonal prisms. It has 14400 vertices, 28800 edges, and 17040 faces (10800 squares, 4800 hexagons, and 1440 decagons). It is the largest nonprismatic convex uniform polychoron.

The vertices and edges form the Cayley graph of the Coxeter group H4.

### Alternate names

• Omnitruncated 120-cell / Omnitruncated 600-cell (Norman W. Johnson)
• Omnitruncated hecatonicosachoron / Omnitruncated hexacosichoron / Omnitruncated polydodecahedron / Omnitruncated polytetrahedron
• Great diprismatohexacosihecatonicosachoron (Acronym gidpixhi) (George Olshevsky, Jonathan Bowers)[4]

### Images

Schlegel diagram (centered on truncated icosidodecahedron) Stereographic projection (Orthogonal view, centered on decagonal prism cell.) (centered on truncated icosidodecahedron) H3 A2/B3 Polychoric net

### Models

The first complete physical model of a 3D projection of the omnitruncated 120-cell was built by a team led by Daniel Duddy and David Richter on August 9, 2006 using the Zome system in the London Knowledge Lab for the 2006 Bridges Conference.[5]

### Full snub 120-cell

Vertex figure for the full snub 120-cell

The full snub 120-cell or full snub 600-cell, defined as an alternation of the omnitruncated 120-cell, can not be made uniform, but it can be given Coxeter diagram , and symmetry [5,3,3]+, and constructed from 1200 octahedrons, 600 icosahedrons, 720 pentagonal antiprisms, 120 snub dodecahedrons, and 7200 tetrahedrons filling the gaps at the deleted vertices. It has 9840 cells, 35040 faces, 32400 edges, and 7200 vertices.[6]

## Related polytopes

These polytopes are a part of a set of 15 uniform polychorons with H4 symmetry:

H4 family polytopes by name, Coxeter diagram, and Schläfli symbol
120-cell rectified
120-cell
truncated
120-cell
cantellated
120-cell
runcinated
120-cell
bitruncated
120-cell
cantitruncated
120-cell
runcitruncated
120-cell
omnitruncated
120-cell
{5,3,3} t1{5,3,3} t0,1{5,3,3} t0,2{5,3,3} t0,3{5,3,3} t1,2{5,3,3} t0,1,2{5,3,3} t0,1,3{5,3,3} t0,1,2,3{5,3,3}
600-cell rectified
600-cell
truncated
600-cell
cantellated
600-cell
runcinated
600-cell
bitruncated
600-cell
cantitruncated
600-cell
runcitruncated
600-cell
omnitruncated
600-cell
{3,3,5} t1{3,3,5} t0,1{3,3,5} t0,2{3,3,5} t0,3{3,3,5} t1,2{3,3,5} t0,1,2{3,3,5} t0,1,3{3,3,5} t0,1,2,3{3,3,5}

## Notes

1. ^ Klitizing, (x3o3o5x - sidpixhi)
2. ^ Klitizing, (x3o3x5x - prix)
3. ^ Klitizing, (x3x3o5x - prahi)
4. ^ Klitizing, (x3x3x5x - gidpixhi)
5. ^ Photos of Zome model of omnitruncated 120/600-cell
6. ^ http://www.bendwavy.org/klitzing/incmats/s3s3s5s.htm